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### XIV Summer Diffiety School

Santo Stefano del Sole (Avellino), Italy.
July 15 - 30, 2011.

This School was organized in cooperation with
and under the scientific direction of Prof. A. M. Vinogradov (Università di Salerno, Italy, and Diffiety Institute, Russia).

### Pictures.

Pictures of XIII Diffiety School may be viewed via the Facebook group "The Diffiety School Experience".

### Courses.

In this edition of the School, the following courses were given.

 09:00-10:45 A1 (L. Vitagliano) B1 (A. Vinogradov) 10:45-11:15 COFFEE BREAK 11:15-13:00 B2 (G. Moreno) A2 (A. Vinogradov)

#### B1. Smooth Manifolds and Observables.

The course aims to show that the natural language of classical physics is differential calculus over commutative algebras and that this fact is a consequence of the classical observability mechanism. From mathematical side this allows to reveal the "logic" of differential calculus. Indeed, differential calculus is the study of certain functors, their representative objects and natural transformations in suitable categories of modules over commutative algebras. On the base of this study all differential geometric concepts may be formalized over an arbitrary commutative algebra. Differential calculus over commutative algebras is not only the "mathematical grammar" of classical nature but it is an indispensable tool in Secondary Calculus.

#### B2. First Order Calculus on Smooth Manifolds.

Lecturer: Giovanni Moreno.
Calculus over smooth manifolds will be developed according to the logic" of Differential Calculus over Commutative Algebras. It will be shown that even in basic differential geometry, this approach reveal more details than the standard one. This course is preparatory to more advanced further courses concerning both the geometry of PDEs and Secondary Calculus.

#### A1: Linear Connections and the Leray-Serre Spectral Sequence.

Lecturer: Luca Vitagliano.
The aim of the course is to introduce the Leray--Serre Spectral Sequence (SS) as the simplest example of $\mathcal{C}$--SS, i.e., the canonical SS associated to a system of PDEs. The course is devided into three parts: in the first part we present fundamentals of the (algebraic) theory of linear connections; in the second part we present a noteworthy example of flat connection, namely the canonical connection in the cohomological bundle associated to a fiber bundle; in the third part we introduce SS and, in particular, a cohomological version of the Leray--Serre one, and prove the celebrated Leray--Serre theorem. If time permits, we will also present fundamentals of Secondary Calculus and the Secondary Calculus on the space of fibers of a bundle as a simple, specific example.

#### A2. Introduction to Geometry of PDEs.

Lecturers: Alexandre M. Vinogrado & Luca Vitagliano
The course will first provide a detailed geometrical analysis of the finite jet spaces, which is indispensable for the theory of diffieties. This is the natural environment in which a system of nonlinear PDEs can be inscribed, and where such concepts like transformations, intrinsic/extrinsic and finite/infinitesimal symmetries, generalized Jacobi brakets, integration by characteristics, Cauchy data, first integrals, etc. become part of the geometry of the PDEs. A survey of solution singularity theory will be given, in connection with the quantization problem.

### Organizing Committee

M. Bächtold, V. Fiore, V. Kalnitsky, G. Moreno, C. Ragano, M. M. Vinogradov, L. Vitagliano.

The Organizing Committee can be contacted via the e-address: